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FIGURE 6.26
One-dimensional case with
m
nodes for the Lagrangian interpolation.
where
N
i
i
are values of
u
at the nodal points. For
Lagrangian interpolation,
N
i
must have the properties
N
i
(ξ
i
)
=
(ξ )
is a polynomial of order
m
−
1, and
v
1
(6.63)
N
i
(ξ
k
)
=
0
(
i
=
k
)
Note that these two characteristics of
N
i
are the same as those of the shape functions defined
in Eq. (6.48).
From the properties of Eq. (6.63),
N
i
(ξ )
can be of the form (
i
term omitted in numerator
and denominator)
m
(ξ
−
ξ
j
)
j
=
1
)
···
(ξ
−
ξ
m
)
(ξ
i
−
ξ
1
)(ξ
i
−
ξ
2
)
···
(ξ
i
−
ξ
i
−
1
)(ξ
i
−
ξ
i
+
1
)
···
(ξ
i
−
ξ
m
)
=
(ξ
−
ξ
)(ξ
−
ξ
)
···
(ξ
−
ξ
i
−
1
)(ξ
−
ξ
i
+
1
j
=
i
1
2
N
i
(ξ )
=
(6.64)
m
(ξ
−
ξ
)
i
j
j
=
1
j
=
i
which are called
Lagrangian polynomials
. As an example, for a 3 node, single DOF per node,
line element, use Eq. (6.62) with
m
=
3 and
=
(ξ
−
ξ
)(ξ
−
ξ
)
=
(ξ
−
ξ
)(ξ
−
ξ
)
=
(ξ
−
ξ
)(ξ
−
ξ
)
2
3
1
3
1
2
N
1
N
2
N
3
(ξ
−
ξ
)(ξ
−
ξ
)
(ξ
−
ξ
)(ξ
−
ξ
)
(ξ
−
ξ
)(ξ
−
ξ
)
1
2
1
3
2
1
2
3
3
1
3
2
Two-Dimensional Case in Cartesian Coordinates
In the two-dimensional case, we seek functions that are uniquely defined at the DOF on
the edge and interior nodes. A simple product of the one-dimensional shape functions for
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